HINT: <no title>
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Use the segment FG⎯⎯⎯⎯⎯⎯⎯⎯
to get two right-angled triangles. Then you can use trigonometric
ratios or the theorem of Pythagoras to work out the answer to the
question.
STEP: Draw a line to create right-angled triangles
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The first thing to do is draw an extra line
across the triangle so that we make two right-angled triangles in the
figure. We do this because we can use the trigonometric ratios and the
theorem of Pythagoras for right-angled triangles.
The line segment GF⎯⎯⎯⎯⎯⎯⎯⎯
in the figure is the line we want: it will create two separate
right-angled triangles! The two right-angled triangles that we get look
like this:
STEP: Use trigonometry to find useful information
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In ΔDGF
(the light blue one) we know one of the non-right angles and one of the
sides. Hence we can use the trigonometric ratios in that triangle
because it is a right-angled triangle.
With the information given, we can find both segments DG⎯⎯⎯⎯⎯⎯⎯⎯⎯ and FG⎯⎯⎯⎯⎯⎯⎯⎯. Remember that we want to get the length of EF⎯⎯⎯⎯⎯⎯⎯⎯, and for that we need both of these lengths. Start by calculating the length of DG⎯⎯⎯⎯⎯⎯⎯⎯⎯, which will allow us to find GE⎯⎯⎯⎯⎯⎯⎯⎯ (because we know that DE⎯⎯⎯⎯⎯⎯⎯⎯=9). This calculation involves the hypotenuse and the side adjacent to D̂ , so use the cosine ratio.
cosθcos(52,4°)(8,9)cos52,4°(8,9)(0,6101...)5,4302...SinceDG⎯⎯⎯⎯⎯⎯⎯⎯⎯+GE⎯⎯⎯⎯⎯⎯⎯⎯=DE⎯⎯⎯⎯⎯⎯⎯⎯:=adjacenthypotenuse=DG⎯⎯⎯⎯⎯⎯⎯⎯⎯8,9=DG⎯⎯⎯⎯⎯⎯⎯⎯⎯=DG⎯⎯⎯⎯⎯⎯⎯⎯⎯=DG⎯⎯⎯⎯⎯⎯⎯⎯⎯GE⎯⎯⎯⎯⎯⎯⎯⎯=9−5,4302...GE⎯⎯⎯⎯⎯⎯⎯⎯=3,5697...
Great: that gets us the value for side GE⎯⎯⎯⎯⎯⎯⎯⎯. Now we need to find the length of side FG⎯⎯⎯⎯⎯⎯⎯⎯. For that we will use the sine ratio (you can also do this calculation with the theorem of Pythagoras, but here we will do it with trigonometry).
sinθsin(52,4°)(8,9)sin52,4°(8,9)(0,7922...)7,0513...=oppositehypotenuse=FG⎯⎯⎯⎯⎯⎯⎯⎯8,9=FG⎯⎯⎯⎯⎯⎯⎯⎯=FG⎯⎯⎯⎯⎯⎯⎯⎯=FG⎯⎯⎯⎯⎯⎯⎯⎯
STEP: Calculate the final answer
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Now we have the lengths of two sides of ΔGEF; since ΔGEF is a right-angled triangle, we can use the theorem of Pythagoras to calculate the length of the hypotenuse.
c2(EF⎯⎯⎯⎯⎯⎯⎯⎯)2∴EF⎯⎯⎯⎯⎯⎯⎯⎯=a2+b2=(FG⎯⎯⎯⎯⎯⎯⎯⎯)2+(GE⎯⎯⎯⎯⎯⎯⎯⎯)2=(7,0513...)2+(3,5697...)2=62,4647...=±62,4647...‾‾‾‾‾‾‾‾‾√←The square root brings inthe ± because a2=(−a)2=±7,9034...≈±7,9
Remember that the instructions say to round the
answer to the first decimal place, as shown in the last step above. Also
notice that we get two different answers, one positive and one
negative. However, the value we calculated represents a distance, so we
must throw out the negative answer.
The final answer is: EF⎯⎯⎯⎯⎯⎯⎯⎯=7,9.
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